Question

If five times the $ 5 ^{th} $ term of an A.P is equal to eight times its $ 8 ^{th} $ term, then show that its $ 13 ^{th} $ term is zero.

Answer 1

Hint: First, we have to know about Arithmetic progression. An arithmetic progression is the sequence of the given numbers and also the difference of any two successive numbers is always a constant.
Here we will find the fifth and eighth terms of the arithmetic progression and then equivalent them after we will solve further to find the $ 13 ^{th} $ term.
That can be expressed using $ a,(a + d),(a + 2d),(a + 3d),... $ where $ a $ is the first term and $ d $ is a common difference.

 Formula used: Formula to consider for solving these questions $ {a_n} = a + (n - 1)d $
Where $ d $ is the common difference, $ a $ is the first term, since we know that difference between consecutive terms is constant in any A.P

Complete step by step answer:
Let from the given that five times the fifth term and also the eight times the eighth term are equal.
This means, by the use of the given arithmetic formula we can make an equation of A.P as $ 5(a + (5 - 1)d) = 8(a + (8 - 1)d) $ where fifth and eighth terms are written in the form of A.P.
Thus, solving this equation, we get $ 5a + 20d = 8a + 56d $ (by multiplying the terms outside and also using the subtraction).
Now we will be simplifying the equation we get $ 8a - 5a + 56d - 20d = 0 $ (tuning all the terms into one side of the equation).
Further solving the above equation, we get $ 8a - 5a + 56d - 20d = 0 \Rightarrow 3a + 36d = 0 $ .
Thus, taking the common terms three and then evaluating to the right side zero we get $ \Rightarrow a + 12d = 0 $ .
Now comparing this term concerning the given A.P that is $ {a_n} = a + (n - 1)d $ .
Hence, we get $ n - 1 = 12 $ (since from the comparison a and d are the same values so they don’t change)
Therefore, we get, $ n - 1 = 12 \Rightarrow n = 13 $ which is the $ 13 ^{th} $ term is zero.

Note: there is also another progression called G.P; the geometric progression is in the form of $ a,ar,a{r^2},... $ where a is the first term and r is the ratio of the G.P.
Since in this given question the main answer is all about finding the terms $ \Rightarrow a + 12d = 0 $ because after finding these we can simply compare it to the general formula to get the required answer.